On the basis of analyzing structure character and state character of epicyclical gear trains (EGTs),topological graph is i mproved. It is realized that gear trains is a physics system,and net problem is presented that is a systems graph theory model applied to solve physics character of EGTs.It is found that topological graph accords with Kirchhoff's law.On the basis of analyzing physics and geometry character of topological graph,branch-characteristics matrix and cut-characteristics matrix are presented,the two are applied to create a systems mathematic model,namely loop-equations and cut-equations.A new fundamental theory is presented for structure characteristics and state characteristics of EGTs based on systems research.And it is applied to velocity,torque and bifurcated power characteristics of EGTs with an example.